Optimal. Leaf size=126 \[ \frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2716,
2721, 2719} \begin {gather*} -\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 2721
Rule 2748
Rubi steps
\begin {align*} \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}+a \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {(3 a) \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {(3 a) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^4 \sqrt {\cos (c+d x)}}\\ &=\frac {2 b}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 70, normalized size = 0.56 \begin {gather*} \frac {4 b-12 a \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+7 a \sin (c+d x)+3 a \sin (3 (c+d x))}{10 d e (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs.
\(2(134)=268\).
time = 10.35, size = 310, normalized size = 2.46
method | result | size |
default | \(-\frac {2 \left (12 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, a \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a -8 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\) | \(310\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 118, normalized size = 0.94 \begin {gather*} \frac {{\left (-3 i \, \sqrt {2} a \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} a \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left ({\left (3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + b\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {7}{2}\right )}}{5 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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